# Correlation between two multivariate time series

I'd like to quantify correlations in time series of 3D position. Assume:

$$p_1(t) = \left( x_1(t), y_1(t), z_1(t) \right) ;$$

$$p_2(t) = \left( x_2(t), y_2(t), z_2(t) \right) .$$

where $x, y,$ and $z$ represent positions as a function of time in a three-dimensional coordinate space. Both time series are equally long. $p$ is constrained by real-world limitations and can be assumed to exhibit a high degree of serial correlation. I can also expect $|p_1(t) - p_2(t)|$ to grow over time.

If I were to plot these in a three-dimensional coordinate space, it would show two separate trajectories over time. These trajectories start at the same point in space and then proceed mostly independently. They might be similar or they might go in complete opposite directions. I need to statistically quantify how similar the two trajectories are. This quantification could be a measure of correlation between the trajectories or some test to establish whether the two multidimensional time series are significantly different.

Does anyone have any recommendations for how this could be done? I am not looking for pairwise correlations of each component (e.g. $\text{corr}(x_1, x_2)$, etc.). I need something that correlates the entire vector (e.g. $\text{corr} (p_1, p_2)$).

• Here are some references I just found by googling: Geiss & Einax "Multivariate correlation analysis - a method for the analysis of multidimensional time series in environmental studies" (1996) and Nguyen et al "Multivariate Maximal Correlation Analysis" (2014). – Richard Hardy Mar 4 '16 at 19:53
• Thanks Richard. I had found the MCA paper but not the paper from '96. I'll take a look and see if it can work. It would be great if the solution had an existing package in R – James Hereford Mar 7 '16 at 15:47
• I checked just now, the link to the 1996 paper still works for me; perhaps you meant that you did not have access to it. Unfortunately, I am not aware of R implementations. – Richard Hardy Mar 7 '16 at 16:01
• @RichardHardy It seems important that the time series are of spatial coordinates. The spatial trajectory of two objects, e.g., is not in the examples in the '96 paper. Not sure why it matters, but seems relevant. – rbatt Jul 25 '16 at 17:33